Prove that if $E \subset \mathbb{R}^n $ has finite perimeter, then almost every vertical slice has finite perimeter too.

Question

Before explaining my problem, I recall the definitions:

Let $E \subset \mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K \subset \mathbb{R}^n$ it holds that \begin{equation*} \label{eq:deflocfiniteperimeter} M_K := \sup \left\{ \int_{E} div\, T(x) \,dx : T \in C^1_c( \mathbb{R}^n; \mathbb{R}^n), spt \,T \subset K, \| T \| \leq 1 \right\} < \infty. \end{equation*} Moreover, if $$ \sup\left\{M_K:K \subset \mathbb{R}^n, K\text{compact}\right\}< \infty,$$ then we say that $E$ is a set of finite perimeter.

Now suppose that $E$ is a set of finite perimeter. I have to prove that for $\mathcal{L}^{n-1}$-a.e. $z \in \mathbb{R}^{n-1}$ the vertical slice $E_z \subset \mathbb{R}$ is a set of finite perimeter; where, by definition, $E_z := \{ t \in \mathbb{R} : (z,t ) \in E\}$.

I can only prove that for a.e. $z \in \mathbb{R}^{n-1}$ the vertical slice is a set of LOCALLY finite perimeter. Here I descrive my attempt:

Let $\rho$ is a regulatizing kernel and consider the sequence $ u_{h} : = \chi_E * \rho_{1/h}$: I already know (from a previous result) that $$ \limsup_{h \to \infty} \int_K |\nabla u_h(x)| dx \leq P(E;K)$$ for all compact sets $K \subset \mathbb{R}^{n}$. Now fix a compact $J \subset \mathbb{R}$. I have proved that if $T \in C^1_c(\mathbb{R})$ satisfies $\|T\| \leq 1$ and $J \supset spt T$, then for a.e. $z \in \mathbb{R}^{n-1}$ it holds $$ \left| \int_{E_z} T'(t) \, dt \right| \leq \liminf_{h \to \infty} \int_{J} |\nabla u_h (z,t)| \, dt .$$ Taking the sup among the functions $T \in C^1_c(\mathbb{R})$ with $\|T\| \leq 1$ and $J \supset spt T$ and integrating on a compact set $H \subset \mathbb{R}^{n-1}$, we get $$ \int_H \sup \left\{ \left| \int_{E_z} T' \right| : T \in C^1_c(\mathbb{R}), \, \|T\| \leq 1 , \, J \supset spt T \right\} \leq \liminf_{h \to \infty} \int_{H \times J} |\nabla u_h| \leq P(E; H \times J ) \leq P(E) < \infty. $$ If the integral of a function is finite then the function is a.e. finite. Hence for a.e. $z$ we have $$ M_{J} := \sup \left\{ \left| \int_{E_z} T' \right| : T \in C^1_c(\mathbb{R}), \, \|T\| \leq 1 , \, J \supset spt T \right\} < \infty.$$ This proves that the set has locally finite perimeter but I can't find a uniform bound for $M_{J}$.

Any help would be really appreciated!

Answer 1

This is exactly the assertion of Vol'pert theorem

Theorem (Vol'pert) Let $E$ a set of finite perimeter in $\mathbb R^n$ with $n>1$ then for $\mathcal L^{n-1}$-a.e. points $x'\in\mathbb R^{n-1}$

• $E_{x'}$ is a set of finite perimeter in $\mathbb R$;
• $\partial^*(E_{x'})=(\partial^* E)_{x'}$;
• $\nu^E_y(x', y)\neq 0$ for every $(x', y)\in\partial ^*(E_{x'})$ where $\nu^E_y$ is the $n$-th component of the generalized inner normal of $E$ named $\nu^E$.

A generalization of this theorem for $k$-dimensional slices where $k=1, 2, \dotsc, n-1$ can be found at [1, Th. 2.93]

[1] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. Oxford: Clarendon Press. xviii, 434 p. (2000). ZBL0957.49001.